PRINCIPAL COMPONENTS AND DIMENSIONALITY

A commonly-asked question in financial modeling refers to the minimum number of stochastic factors that must be included in the model in order to properly capture the price and risk behavior of financial instruments. The number of stochastic components needed to describe an instrument (or portfolio of instruments) to a desired level of accuracy is referred to as the dimensionality of the instrument. It is commonly observed that the U.S. dollar term structure can be described satisfactorily with at most three factors. What you are most often interested in, however, is not in the number of stochastic factors you need to describe the term structure itself, but in the number of stochastic components needed to price and risk manage instruments that depend on the term structure. The dimensionality of instruments, such as derivatives, that depend on the term structure of interest rates is not the same as the dimensionality of the term structure itself. This article attempts to clarify this concept and presents a technique to determine the dimensionality of financial instruments.

The number of stochastic components needed to describe a financial instrument is embodied in the concept of principal components (PC). Given a set of correlated random variables, X ={x_i}, the PC of X are another set of random variables, Y ={y_i}, with the following properties:

1. The y_i are uncorrelated

2. Eachy_i is a linear combination of the x_i

3. The sample variance of each y_i is as large as possible

(subject to certain constraints)

4. The y_i are ordered such that their sample variance

decreases as _i increases

5. The total variance of the principal components equals the

total variance of the original variables. The j^th principal

component is:

The coefficients of this linear combination have the following properties:

1. The j^th row of coefficients a_ji is the eigenvector

corresponding to the j^th largest eigenvalue of the

covariance matrix of the {x_i}

2. The variance of y_j equals the j^th eigenvalue of the

covariance matrix

Since the PC are independent, the total variance of the PC equals the sum of the eigenvalues of the covariance matrix. If *_j denotes the j^th eigenvalue of the covariance matrix of {x_i}, the fraction of variability explained by y_j is as follows:

proportion of variance explained by 

To apply the concept of PC to financial instruments, consider a description of the term structure of interest rates in terms of constant maturity forward rates. Let f_i (t_j) be the forward rate observed at timet_j and applicable to the period starting at T_i + t_j and ending at T_i+₁+ t_j. In the case of the LIBOR term structure, for example, T_i and T_i+1 are the maturities of two consecutive LIBOR instruments. Consider now an instrument (or portfolio of instruments), whose price at time t_j is V(tj). For simplicity, assume that the price of the instrument is only a function of {f _i (t_j)}= {f ₁(t_j), f ₂(t_j), .... , f_n(t_j)}.

We are interested in the dimensionality of the relative changes in the value of the instrument. By Ito's lemma, the changes in value of the instrument at time t_j, dV(t_j) /V(t_j), are related to the changes in forward rates, df_i (t_j), by the expression:

The drift µ(t_j) is a locally deterministic function and the df_i(t_j) are serially uncorrelated (the function µ(t_j) depends on the value of the instrument, of the time derivative of the value, of its first and second derivatives with respect to the forward rates, and of the covariances of the forward rates.) If we interpret f_i(t_j) as historically observed rates, and if we assume that our instrument is rolled over at every t_i, the variance of value changes is given by the following expression:

Here *_i(t_j) is the delta of the instrument with respect to f_i.

To implement this analysis, you proceed as follows. Given a time series of forward rates observations, you compute the price of the instrument and the vector of deltas with respect to forward rates. With this information, you construct a set of time series:

There will be as many time series as there are forward rates (notice that the drift does not enter into the analysis.) Using a standard statistical analysis package (such as SAS or SPLUS), you now perform a PC analysis on these time series.

As described, the PC analysis is based on the covariance matrix of the dependent random variables. The analysis can also be conducted based on the correlation matrix. Using the correlation matrix, however, artificially evens out the importance of the individual variability of the original variables and therefore tends to lend less explainatory power to the most importent principal components.

To illustrate the use of this technique, consider two simple cases. The first case is the U.S. dollar LIBOR term structure itself, described by LIBOR deposits and swaps up to 10 years (eight basic securities were used). The second case is a European swaption struck on a semi-annual swap commencing in 30 days with final payment in 10 years (the values of implied volatility and other pricing details are not relevant to these observations.) As the figure shows, three principal components are sufficient to explain about 87% of the variance of the term structure, while only two principal components are enough to explain over 90% of the variance of the swaption. This simple example clearly shows that the dimensionality of the swaption is much smaller than that of the term structure.

This article was written by Susan Chiu, assistant v.p. with the portfolio strategies group at Wells Fargo Bank, San Francisco.